#7542004

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1783 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(5,6)(7,10)(8,12)(9,11)(13,14,19,24)(15,25,27,22)(16,21,28,36)(17,20,31,35)(18,32,33,23)(26,39,40,37)(29,34,41,44)(30,43,42,38);(1,2)(3,4)(5,7,9,12)(6,10,11,8)(13,15,18,17,19,27,33,31)(14,20,23,25,24,35,32,22)(16,26,30,29,28,40,42,41)(21,34,38,39,36,44,43,37);(5,8,9,10)(6,7,11,12)(13,16,19,28)(14,21,24,36)(15,26,27,40)(17,29,31,41)(18,30,33,42)(20,34,35,44)(22,37,25,39)(23,38,32,43);(2,4)(6,11)(7,12)(15,27)(16,28)(17,31)(20,35)(21,36)(22,25)(30,42)(38,43);(13,17,18,27,19,31,33,15)(14,22,23,20,24,25,32,35)(16,29,30,40,28,41,42,26)(21,37,38,34,36,39,43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=gi,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7449200);